Optimality Conditions for Variational Problems in Incomplete Functional Spaces

This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our approach consists of reducing a variational problem to a (nondynamic) problem of constrained optimization in...

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Veröffentlicht in:	Journal of optimization theory and applications Jg. 193; H. 1-3; S. 139 - 157
Hauptverfasser:	Mohammadi, Ashkan, Mordukhovich, Boris S.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.06.2022 Springer Nature B.V
Schlagworte:	Applications of Mathematics Calculus Calculus of variations Calculus of Variations and Optimal Control; Optimization Constraints Continuity (mathematics) Engineering Mathematical functions Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Subspaces Theory of Computation Velocity Variational analysis Necessary optimality conditions 49J53 49J52 Optimal control 90C48 Calculus of variations Generalized differentiation Constrained optimization 49K24
ISSN:	0022-3239, 1573-2878
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Abstract	This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our approach consists of reducing a variational problem to a (nondynamic) problem of constrained optimization in a normed space and then applying the results recently obtained for the latter class by using generalized differentiation. In this way, we derive necessary optimality conditions for nonconvex problems of the calculus of variations with velocity constraints under the weakest metric subregularity-type constraint qualification. The developed approach leads us to a short and simple proof of first-order necessary optimality conditions for such and related problems in broad spaces of functions including those of class C k as k ≥ 1 .
AbstractList	This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our approach consists of reducing a variational problem to a (nondynamic) problem of constrained optimization in a normed space and then applying the results recently obtained for the latter class by using generalized differentiation. In this way, we derive necessary optimality conditions for nonconvex problems of the calculus of variations with velocity constraints under the weakest metric subregularity-type constraint qualification. The developed approach leads us to a short and simple proof of first-order necessary optimality conditions for such and related problems in broad spaces of functions including those of class Ck as k≥1. This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our approach consists of reducing a variational problem to a (nondynamic) problem of constrained optimization in a normed space and then applying the results recently obtained for the latter class by using generalized differentiation. In this way, we derive necessary optimality conditions for nonconvex problems of the calculus of variations with velocity constraints under the weakest metric subregularity-type constraint qualification. The developed approach leads us to a short and simple proof of first-order necessary optimality conditions for such and related problems in broad spaces of functions including those of class C k as k ≥ 1 .
Author	Mohammadi, Ashkan Mordukhovich, Boris S.
Author_xml	– sequence: 1 givenname: Ashkan surname: Mohammadi fullname: Mohammadi, Ashkan organization: Department of Mathematics and Statistics, Georgetown University – sequence: 2 givenname: Boris S. surname: Mordukhovich fullname: Mordukhovich, Boris S. email: boris@math.wayne.edu organization: Department of Mathematics, Wayne State University
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Cites_doi	10.1007/978-1-4471-4820-3 10.1287/moor.2020.1074 10.1287/moor.2016.0789 10.1137/15M1012608 10.1090/S0002-9947-1993-1156300-4 10.1137/20M1342215 10.1137/100813415 10.1090/tran/8253 10.1007/978-3-642-02431-3 10.1137/0806015 10.1007/978-3-642-69512-4 10.1002/mana.19921550110 10.1007/978-3-319-64277-2
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Keywords	Variational analysis Necessary optimality conditions 49J53 49J52 Optimal control 90C48 Calculus of variations Generalized differentiation Constrained optimization 49K24
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References	GfrererHOutrataJVOn computation of generalized derivatives of the normal-cone mapping and their applicationsMath. Oper. Res.20164115351556369374010.1287/moor.2016.0789 MohammadiAMordukhovichBSSarabiMEVariational analysis of composite models with applications to continuous optimizationMath. Oper. Res. (To Appear)202110.1287/moor.2020.1074 MordukhovichBSVariational Analysis and Generalized Diffrerentiation, I: Basic Theory, II: Applications2006BerlinSpringer GfrererHMordukhovichBSComplete characterizations of tilt stability in nonlinear programming under weakest qualification conditionsSIAM J. Optim.20152520812119341359610.1137/15M1012608 Tonelli, L.: 1923) Fondamenti di Calcolo delle Variazoni. I. Bologna, II, Nicola Zanichelli (1921 GfrererHFirst-order and second-order characterizations of metric subregularity and calmness of constraint set mappingsSIAM J. Optim.20112114391474286950410.1137/100813415 ClarkeFFunctional Analysis. Calculus of Variations and Optimal Control2013LondonSpringer10.1007/978-1-4471-4820-3 GoldbergHKampowskyWTroltzschFOn Nemytskij operators in LP spaces of abstract functionsMath. Nachr.1992155127140123126010.1002/mana.19921550110 VinterRBOptimal Control2000Boston, MABirkhäuser0952.49001 AubinJ-PCellinaADifferential Inclusions1984BerlinSpringer10.1007/978-3-642-69512-4 SmirnovGVIntroduction to the Theory of Differential Inclusions2002Providence, RIAmerican Mathematical Society0992.34001 BurkeJVTsengPUnified analysis of Hoffmans bound via Fenchel dualitySIAM J. Optim.19966265282138732510.1137/0806015 IoffeADVariational Analysis of Regular Mappings: Theory and Applications2017Cham, SwitzerlandSpringer10.1007/978-3-319-64277-2 BolzaOLectures on the Calculus of Variations1904Chicago, ILThe University of Chicago Press35.0373.01 MordukhovichBSComplete characterizations of openness, metric regularity, and Lipschitzian properties of multifunctionsTrans. Am. Math. Soc.1993340135115630010.1090/S0002-9947-1993-1156300-4 RockafellarRTWetsRJ-BVariational Analysis1998BerlinSpringer10.1007/978-3-642-02431-3 MohammadiAMordukhovichBSVariational analysis in normed spaces with applications in constraint optimizationSIAM J. Optim.202131569603421571810.1137/20M1342215 MohammadiAMordukhovichBSSarabiMEParabolic regularity in geometric variatiional analysisTrans. Am. Math. Soc.20213741711176310.1090/tran/8253 BlissGALectures on the Calculus of Variations1946Chicago, ILThe University of Chicago Press0063.00459 PontryaginLSBoltyanskiiVGGamkrelidzeRVMishchenkoRFThe Mathematical Theory of Optimal Processes1962New YorkWiley0102.32001 H Gfrerer (1964_CR7) 2015; 25 RB Vinter (1964_CR20) 2000 AD Ioffe (1964_CR10) 2017 BS Mordukhovich (1964_CR15) 2006 O Bolza (1964_CR3) 1904 J-P Aubin (1964_CR1) 1984 H Gfrerer (1964_CR8) 2016; 41 RT Rockafellar (1964_CR17) 1998 GV Smirnov (1964_CR18) 2002 A Mohammadi (1964_CR13) 2021 LS Pontryagin (1964_CR16) 1962 1964_CR19 F Clarke (1964_CR5) 2013 H Gfrerer (1964_CR6) 2011; 21 A Mohammadi (1964_CR11) 2021; 31 BS Mordukhovich (1964_CR14) 1993; 340 H Goldberg (1964_CR9) 1992; 155 GA Bliss (1964_CR2) 1946 JV Burke (1964_CR4) 1996; 6 A Mohammadi (1964_CR12) 2021; 374
References_xml	– reference: GoldbergHKampowskyWTroltzschFOn Nemytskij operators in LP spaces of abstract functionsMath. Nachr.1992155127140123126010.1002/mana.19921550110 – reference: MohammadiAMordukhovichBSVariational analysis in normed spaces with applications in constraint optimizationSIAM J. Optim.202131569603421571810.1137/20M1342215 – reference: SmirnovGVIntroduction to the Theory of Differential Inclusions2002Providence, RIAmerican Mathematical Society0992.34001 – reference: VinterRBOptimal Control2000Boston, MABirkhäuser0952.49001 – reference: BurkeJVTsengPUnified analysis of Hoffmans bound via Fenchel dualitySIAM J. Optim.19966265282138732510.1137/0806015 – reference: ClarkeFFunctional Analysis. Calculus of Variations and Optimal Control2013LondonSpringer10.1007/978-1-4471-4820-3 – reference: BlissGALectures on the Calculus of Variations1946Chicago, ILThe University of Chicago Press0063.00459 – reference: Tonelli, L.: 1923) Fondamenti di Calcolo delle Variazoni. I. Bologna, II, Nicola Zanichelli (1921) – reference: AubinJ-PCellinaADifferential Inclusions1984BerlinSpringer10.1007/978-3-642-69512-4 – reference: IoffeADVariational Analysis of Regular Mappings: Theory and Applications2017Cham, SwitzerlandSpringer10.1007/978-3-319-64277-2 – reference: BolzaOLectures on the Calculus of Variations1904Chicago, ILThe University of Chicago Press35.0373.01 – reference: MohammadiAMordukhovichBSSarabiMEVariational analysis of composite models with applications to continuous optimizationMath. Oper. Res. (To Appear)202110.1287/moor.2020.1074 – reference: RockafellarRTWetsRJ-BVariational Analysis1998BerlinSpringer10.1007/978-3-642-02431-3 – reference: MordukhovichBSComplete characterizations of openness, metric regularity, and Lipschitzian properties of multifunctionsTrans. Am. Math. Soc.1993340135115630010.1090/S0002-9947-1993-1156300-4 – reference: GfrererHOutrataJVOn computation of generalized derivatives of the normal-cone mapping and their applicationsMath. Oper. Res.20164115351556369374010.1287/moor.2016.0789 – reference: GfrererHMordukhovichBSComplete characterizations of tilt stability in nonlinear programming under weakest qualification conditionsSIAM J. Optim.20152520812119341359610.1137/15M1012608 – reference: MohammadiAMordukhovichBSSarabiMEParabolic regularity in geometric variatiional analysisTrans. Am. Math. Soc.20213741711176310.1090/tran/8253 – reference: PontryaginLSBoltyanskiiVGGamkrelidzeRVMishchenkoRFThe Mathematical Theory of Optimal Processes1962New YorkWiley0102.32001 – reference: GfrererHFirst-order and second-order characterizations of metric subregularity and calmness of constraint set mappingsSIAM J. Optim.20112114391474286950410.1137/100813415 – reference: MordukhovichBSVariational Analysis and Generalized Diffrerentiation, I: Basic Theory, II: Applications2006BerlinSpringer – volume-title: Functional Analysis. Calculus of Variations and Optimal Control year: 2013 ident: 1964_CR5 doi: 10.1007/978-1-4471-4820-3 – ident: 1964_CR19 – volume-title: Lectures on the Calculus of Variations year: 1904 ident: 1964_CR3 – year: 2021 ident: 1964_CR13 publication-title: Math. Oper. Res. (To Appear) doi: 10.1287/moor.2020.1074 – volume: 41 start-page: 1535 year: 2016 ident: 1964_CR8 publication-title: Math. Oper. Res. doi: 10.1287/moor.2016.0789 – volume: 25 start-page: 2081 year: 2015 ident: 1964_CR7 publication-title: SIAM J. Optim. doi: 10.1137/15M1012608 – volume-title: The Mathematical Theory of Optimal Processes year: 1962 ident: 1964_CR16 – volume: 340 start-page: 1 year: 1993 ident: 1964_CR14 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1993-1156300-4 – volume-title: Lectures on the Calculus of Variations year: 1946 ident: 1964_CR2 – volume: 31 start-page: 569 year: 2021 ident: 1964_CR11 publication-title: SIAM J. Optim. doi: 10.1137/20M1342215 – volume: 21 start-page: 1439 year: 2011 ident: 1964_CR6 publication-title: SIAM J. Optim. doi: 10.1137/100813415 – volume: 374 start-page: 1711 year: 2021 ident: 1964_CR12 publication-title: Trans. Am. Math. Soc. doi: 10.1090/tran/8253 – volume-title: Variational Analysis year: 1998 ident: 1964_CR17 doi: 10.1007/978-3-642-02431-3 – volume: 6 start-page: 265 year: 1996 ident: 1964_CR4 publication-title: SIAM J. Optim. doi: 10.1137/0806015 – volume-title: Differential Inclusions year: 1984 ident: 1964_CR1 doi: 10.1007/978-3-642-69512-4 – volume-title: Optimal Control year: 2000 ident: 1964_CR20 – volume-title: Variational Analysis and Generalized Diffrerentiation, I: Basic Theory, II: Applications year: 2006 ident: 1964_CR15 – volume-title: Introduction to the Theory of Differential Inclusions year: 2002 ident: 1964_CR18 – volume: 155 start-page: 127 year: 1992 ident: 1964_CR9 publication-title: Math. Nachr. doi: 10.1002/mana.19921550110 – volume-title: Variational Analysis of Regular Mappings: Theory and Applications year: 2017 ident: 1964_CR10 doi: 10.1007/978-3-319-64277-2
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SubjectTerms	Applications of Mathematics Calculus Calculus of variations Calculus of Variations and Optimal Control; Optimization Constraints Continuity (mathematics) Engineering Mathematical functions Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Subspaces Theory of Computation Velocity
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Title	Optimality Conditions for Variational Problems in Incomplete Functional Spaces
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